The Slepian model based independent interval approximation of persistency and zero-level excursion distributions

Abstract

In physics and engineering literature, the distribution of the excursion time of a stationary Gaussian process has been approximated through a method based on a stationary switch process with independently distributed switching times. The approach matches the covariance of the clipped Gaussian process with that of the stationary switch process. By expressing the switching time distribution as a function of the covariance, the so-called independent interval approximation (IIA) is obtained for the excursions of Gaussian processes. This approach has successfully approximated the persistency coefficient for many vital processes in physics but left an unanswered question about when such an approach leads to a mathematically meaningful and proper excursion distribution. Here, we propose an alternative approximation: the Slepian-based IIA. This approach matches the expected values of the clipped Slepian process and the corresponding switch process initiated at the origin. It is shown that these two approaches are equivalent, and thus, the original question of the mathematical validity of the IIA method can be rephrased using the Slepian model setup. We show that this approach leads to valid approximations of the excursion distribution for a large subclass of the Gaussian processes with monotonic covariance. Within this class, the approximated excursion time distribution has a stochastic representation that connects directly to the covariance of the underlying Gaussian process. This representation is then used to approximate the persistency coefficient for several important processes to illustrate the Slepian-based IIA approach. Lastly, we argue that the ordinary IIA is ill-suited in certain situations, such as for Gaussian processes with a non-monotonic covariance.